Journées de l'optimisation 2007 Optimizations days

Convex optimization was radically transformed following Fenchel's work (1951), and was articulated by Rockafellar in his classic text (1970) and in the more recent variational/nonsmooth analysis text by Rockafellar and Wets (1998). Yet, some of the fundamental underpinnings of the field have remained poorly understood by a broad audience, because of the difficulty of the analysis and the lack of unification. This talk will review an effort by the author in a recent book to restructure the subject using a handful of unifying principles that can be easily visualized and readily understood. Several new lines of research and analysis were necessary for this, including: (1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. (2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. (3) A unification of the major constraint qualifications that guarantee the existence of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions.